Cohabiting couples are for the first time as likely to have children as those who marry, according to a report on changing family relationships in the UK published on Thursday.

Cohabitation is increasingly seen as an alternative for long-term relationships and raising children, rather than as a "trial run" before marriage, the study commissioned by the Co-operative's new family law service suggests.

A significant proportion of those who have not married, however, are unaware of their rights because the law has failed to keep up with social change, Christina Blacklaws, of the Co-operative Legal Services, warns.

The evidence of shifting parenting patterns emerges in data extracted from the 2001 and 2011 censuses. It has been assembled by Dr Esmée Hanna and Dr David Grainger, both of Leeds University, who specialise in family relationship changes over time.

Their research reveals a 34% increase in the number of cohabiting couples with dependent children in the 10 years to 2011. Over that period, the total number of cohabiting families with dependent children – those aged under 18 – increased by 292,000, while the number of married couples with dependent children fell by 319,000.

Last year, 38% of cohabiting couples were parents – the same percentage as married couples with children. Of live births in 2010, 31% were to women cohabiting with but not married to their partner – up from 25% in 2001. There are still more married couples with children than cohabiting couples with children in the UK.

The rapid changes are reflected in a survey commissioned for the report showing that 52% believe marriage is not important providing that parents are in a committed relationship. Only 27%, according to the survey of more than 2,000 adults carried out by YouGov, take the more traditional view that couples should be married before having children.

Despite increasing social acceptance, the report acknowledges that cohabiting couple families continue to be less stable than married couple families. A higher proportion of all family breakdowns involve young children from unmarried parents, the research reveals.

There is also substantial confusion over the legal rights of cohabiting couples as compared to married couples. Of adults questioned, over a quarter, 26%, believe cohabiting couples have the same rights as married couples when it comes to child custody, 22% when it comes to property and 21% when it comes to finances.

Hanna and Grainger commented: "Part of the decline in marriage in recent years can perhaps be explained by the increase in cohabitation as a family format, with ever more couples choosing to live together. And as the number of cohabiting couples increases, they are increasingly seen as a socially legitimate family environment for childbearing."

Blacklaws, who is director of family law at the Co-operative Legal Services, said: "Although many people still believe they have rights as common-law spouses, there is no such status in law. As a result, some cohabiting families may find themselves facing real difficulties should they split up, particularly when there are children involved.

"It is clear that this area of family law is in urgent need of an overhaul. However, in the meantime, people need to think carefully about how they protect themselves and their families – preferably by reaching and signing agreements about what would happen if you did split up. This could save a huge amount of cost and heartache if the worst happens."